A power law for cells

The human cell count and size distribution

Cell size and cell count are adaptively regulated and intimately linked to growth and function. Yet, despite their widespread relevance, the relation between cell size and count has never been formally examined over the whole human body. Here, we compile a comprehensive dataset of cell size and count over all major cell types, with data drawn from >1,500 published sources. We consider the body of a representative male (70 kg), which allows further estimates of a female (60 kg) and 10-y-old child (32 kg). We build a hierarchical interface for the cellular organization of the body, giving easy access to data, methods, and sources (https://humancelltreemap.mis.mpg.de/). In total, we estimate total body counts of ≈36 trillion cells in the male, ≈28 trillion in the female, and ≈17 trillion in the child. These data reveal a surprising inverse relation between cell size and count, implying a trade-off between these variables, such that all cells within a given logarithmic size class contribute an equal fraction to the body's total cellular biomass. We also find that the coefficient of variation is approximately independent of mean cell size, implying the existence of cell-size regulation across cell types. Our data serve to establish a holistic quantitative framework for the cells of the human body, and highlight large-scale patterns in cell biology.

Variance in Landscape Connectivity Shifts Microbial Population Scaling

Understanding mechanisms shaping distributions and interactions of soil microbes is essential for determining their impact on large scale ecosystem services, such as carbon sequestration, climate regulation, waste decomposition, and nutrient cycling. As the functional unit of soil ecosystems, we focus our attention on the spatial structure of soil macroaggregates. Emulating this complex physico-chemical environment as a patchy habitat landscape we investigate on-chip the effect of changing the connectivity features of this landscape as Escherichia coli forms a metapopulation. We analyze the distributions of E. coli occupancy using Taylor's law, an empirical law in ecology which asserts that the fluctuations in populations is a power law function of the mean. We provide experimental evidence that bacterial metapopulations in patchy habitat landscapes on microchips follow this law. Furthermore, we find that increased variance of patch-corridor connectivity leads to a qualitative transition in the fluctuation scaling. We discuss these results in the context of the spatial ecology of microbes in soil.

Cell type innovation at the tips of the animal tree

Understanding how organs originate is challenging due to the twin problems of explaining how new cell types evolve and how collective interactions between cell types arise and become selectively advantageous. Animals are assemblages of organs and cell types of different antiquities, and among the most rapidly and convergently evolving are exocrine glands and their constituent secretory cell types. Such structures have arisen independently thousands of times across the Metazoa, impacting how animals chemically interact with their environments. The recurrent evolution of exocrine systems provides a paradigm for examining how qualitative phenotypic novelties arise from variation at the cellular level. Here, we take a hierarchical perspective, focusing on the evolutionary assembly of novel biosynthetic pathways and secretory cell types, and how both selection and non-adaptive molecular processes may combine to build the complex, modular architectures of many animal glands.

Neurodegeneration-related beta-amyloid as autocatabolism-attenuator in a micro-in vivo system

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Bdelloids are adaptive models for studying aggregate-metabolism interactions.

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Starvation causes reversible organ shrinkage in bdelloids.

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The organ shrinkage is in connection with autocatabolic processes.

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Beta-amyloid attenuates the starvation-induced germovitellaria shrinkage.

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Human-type amyloid-aggregates are metabolism-regulators in two bdelloid species.

Investigation of human neurodegeneration-related aggregates of beta-amyloid 1–42 (Aβ42) on bdelloid rotifers is a novel interdisciplinary approach in life sciences. We reapplied an organ size-based in vivo monitoring system, exploring the autocatabolism-related alterations evoked by Aβ42, in a glucose-supplemented starvation model. The experientially easy-to-follow size reduction of the bilateral reproductive organ (germovitellaria) in fasted rotifers was rescued by Aβ42, serving as a nutrient source- and peptide sequence-specific attenuator of the organ shrinkage phase and enhancer of the regenerative one including egg reproduction. Recovery of the germovitellaria was significant in comparison with the greatly shrunken form. In contrast to the well-known neurotoxic Aβ42 (except the bdelloids) with specific regulatory roles, the artificially designed scrambled version (random order of amino acids) was inefficient in autocatabolism attenuation, behaving as negative control. This native Aβ42-related modulation of the ‘functionally reversible organ shrinkage’ can be a potential experiential and supramolecular marker of autocatabolism in vivo.

Macroecological laws describe variation and diversity in microbial communities

How the coexistence of many species is maintained is a fundamental and unresolved question in ecology. Coexistence is a puzzle because we lack a mechanistic understanding of the variation in species presence and abundance. Whether variation in ecological communities is driven by deterministic or random processes is one of the most controversial issues in ecology. Here, I study the variation of species presence and abundance in microbial communities from a macroecological standpoint. I identify three macroecological laws that quantitatively characterize the fluctuation of species abundance across communities and over time. Using these three laws, one can predict species' presence and absence, diversity, and commonly studied macroecological patterns. I show that a mathematical model based on environmental stochasticity, the stochastic logistic model, quantitatively predicts the three macroecological laws, as well as non-stationary properties of community dynamics.

Raising the Connectome: The Emergence of Neuronal Activity and Behavior in Caenorhabditis elegans

The differentiation of neurons and formation of connections between cells is the basis of both the adult phenotype and behaviors tied to cognition, perception, reproduction, and survival. Such behaviors are associated with local (circuits) and global (connectome) brain networks. A solid understanding of how these networks emerge is critical. This opinion piece features a guided tour of early developmental events in the emerging connectome, which is crucial to a new view on the connectogenetic process. Connectogenesis includes associating cell identities with broader functional and developmental relationships. During this process, the transition from developmental cells to terminally differentiated cells is defined by an accumulation of traits that ultimately results in neuronal-driven behavior. The well-characterized developmental and cell biology of Caenorhabditis elegans will be used to build a synthesis of developmental events that result in a functioning connectome. Specifically, our view of connectogenesis enables a first-mover model of synaptic connectivity to be demonstrated using data representing larval synaptogenesis. In a first-mover model of Stackelberg competition, potential pre- and postsynaptic relationships are shown to yield various strategies for establishing various types of synaptic connections. By comparing these results to what is known regarding principles for establishing complex network connectivity, these strategies are generalizable to other species and developmental systems. In conclusion, we will discuss the broader implications of this approach, as what is presented here informs an understanding of behavioral emergence and the ability to simulate related biological phenomena.

Tracking Poisson Parameter for Non-Stationary Discontinuous Time Series with Taylor’s Abnormal Fluctuation Scaling

We propose a parameter estimation method for non-stationary Poisson time series with the abnormal fluctuation scaling, known as Taylor’s law. By introducing the effect of Taylor’s fluctuation scaling into the State Space Model with the Particle Filter, the underlying Poisson parameter’s time evolution is estimated correctly from given non-stationary time series data with abnormally large fluctuations. We also developed a discontinuity detection method which enables tracking the Poisson parameter even for time series including sudden discontinuous jumps. As an example of application of this new general method, we analyzed Point-of-Sales data in convenience stores to estimate change of probability of purchase of commodities under fluctuating number of potential customers. The effectiveness of our method for Poisson time series with non-stationarity, large discontinuities and Taylor’s fluctuation scaling is verified by artificial and actual time series.

The lens growth process

The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at its surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.

A full lifespan model of vertebrate lens growth

The mathematical determinants of vertebrate organ growth have yet to be elucidated fully. Here, we utilized empirical measurements and a dynamic branching process-based model to examine the growth of a simple organ system, the mouse lens, from E14.5 until the end of life. Our stochastic model used difference equations to model immigration and emigration between zones of the lens epithelium and included some deterministic elements, such as cellular footprint area. We found that the epithelial cell cycle was shortened significantly in the embryo, facilitating the rapid growth that marks early lens development. As development progressed, epithelial cell division becomes non-uniform and four zones, each with a characteristic proliferation rate, could be discerned. Adjustment of two model parameters, proliferation rate and rate of change in cellular footprint area, was sufficient to specify all growth trajectories. Modelling suggested that the direction of cellular migration across zonal boundaries was sensitive to footprint area, a phenomenon that may isolate specific cell populations. Model runs consisted of more than 1000 iterations, in each of which the stochastic behaviour of thousands of cells was followed. Nevertheless, sequential runs were almost superimposable. This remarkable degree of precision was attributed, in part, to the presence of non-mitotic flanking regions, which constituted a path by which epithelial cells could escape the growth process. Spatial modelling suggested that clonal clusters of about 50 cells are produced during migration and that transit times lengthen significantly at later stages, findings with implications for the formation of certain types of cataract.

Property of Fluctuations of Sales Quantities by Product Category in Convenience Stores

The ability to ascertain the extent of product sale fluctuations for each store and locality is indispensable to inventory management. This study analyzed POS data from 158 convenience stores in Kawasaki City, Kanagawa Prefecture, Japan and found a power scaling law between the mean and standard deviation of product sales quantities for several product categories. For the statistical domains of low sales quantities, the power index was 1/2; for large sales quantities, the power index was 1, so called Taylor’s law holds. The value of sales quantities with changing power indixes differed according to product category. We derived a Poissonian compound distribution model taking into account fluctuations in customer numbers to show that the scaling law could be explained theoretically for most of items. We also examined why the scaling law did not hold in some exceptional cases.

A Process-Independent Explanation for the General Form of Taylor's Law

Taylor's law (TL) describes the scaling relationship between the mean and variance of populations as a power law. TL is widely observed in ecological systems across space and time, with exponents varying largely between 1 and 2. Many ecological explanations have been proposed for TL, but it is also commonly observed outside ecology. We propose that TL arises from the constraining influence of two primary variables: the number of individuals and the number of censuses or sites. We show that most possible configurations of individuals among censuses or sites produce the power-law form of TL, with exponents between 1 and 2. This "feasible set" approach suggests that TL is a statistical pattern driven by two constraints, providing an a priori explanation for this ubiquitous pattern. However, the exact form of any specific mean-variance relationship cannot be predicted in this way, that is, this approach does a poor job of predicting variation in the exponent, suggesting that TL may still contain ecological information.

Optimality conditions for cell-fate heterogeneity that maximize the effects of growth factors in PC12 cells

Recently, the heterogeneity that arises from stochastic fate decisions has been reported for several types of cancer-derived cell lines and several types of clonal cells grown under constant environmental conditions. However, the relation between this stochasticity and the responsiveness to extracellular stimuli remains largely unknown. Here we focused on the fate decisions of the PC12 cell line, which was derived from rat pheochromocytoma, and is a model system to study differentiation into sympathetic neurons. Whereas epidermal growth factor (EGF) stimulates the proliferation of populations of PC12 cells, nerve growth factor (NGF) promotes the differentiation of neurites to neuron-like cells. We found that phenotypic heterogeneity increased with time at several surrounding serum concentrations, suggesting stochastic cell-fate decisions in single cells. We made a simple mathematical model assuming Markovian transitions of the cell fates, and estimated the transition rates based on Bayes' theorem. The model suggests that depending on the serum concentration, EGF (NGF) even directs differentiation (proliferation) at the single-cell level. The maximum effects of the growth factors were ensured when the transition rates were appropriately controlled by the serum concentration to produce a nonextremal, moderate amount of cell-fate heterogeneity. Our model was validated by the experimental finding that the means and variances of the local cell densities obey a power-law relationship. These results suggest that even when efficient responses to growth factors are observed at the population level, the growth factors stochastically direct the cell-fate decisions in different directions at the single-cell level.

Elucidation of the mechanisms that regulate cell fate has become one of the primary goals of research in cell biology and regenerative medicine. Growth factors are often used to regulate cell fate. However, stochastic cellular responses to growth regulators have prevented precise control of cell fate. We report our investigation of the relationship between heterogeneity and responsiveness in cell fate decisions by both single cells and populations of cells. Our study involved PC12, a cultured cell line for which cell-fates are affected by exposure to growth factors and culture conditions. Computational methods using a mathematical model enabled us to determine the cell-fate decisions rate in single PC12 cells and analyze the population responses to growth factors from experimental data. Our findings reveal that growth factors control cell-fate decisions rate in single PC12 cells, and suggest distinct differences in the mechanisms of actions of growth factors under different culture conditions. In addition, we observed maximum effects of growth factors when a nonextremal, moderate amount of cell-fate heterogeneity exists. Our results give several insights into stochastic cell responses, including the effects of anticancer agents on cancer cells and the optimization of methods to induce the differentiation of stem cells.

Stochastic multiplicative population growth predicts and interprets Taylor's power law of fluctuation scaling

Taylor's law (TL) asserts that the variance of the density (individuals per area or volume) of a set of comparable populations is a power-law function of the mean density of those populations. Despite the empirical confirmation of TL in hundreds of species, there is little consensus about why TL is so widely observed and how its estimated parameters should be interpreted. Here, we report that the Lewontin-Cohen (henceforth LC) model of stochastic population dynamics, which has been widely discussed and applied, leads to a spatial TL in the limit of large time and provides an explicit, exact interpretation of its parameters. The exponent of TL exceeds 2 if and only if the LC model is supercritical (growing on average), equals 2 if and only if the LC model is deterministic, and is less than 2 if and only if the LC model is subcritical (declining on average). TL and the LC model describe the spatial variability and the temporal dynamics of populations of trees on long-term plots censused over 75 years at the Black Rock Forest, Cornwall, NY, USA.

Distributions of reproductive and somatic cell numbers in diverse Volvox (Chlorophyta) species

BACKGROUND

Volvox (Chlorophyta) asexual colonies consist of two kinds of cells: a large number of small somatic cells and a few large reproductive cells. The numbers of reproductive and somatic cells correspond directly to the major components of fitness - fecundity and viability, respectively. Volvox species display diverse patterns of development that give rise to the two cell types.

QUESTIONS

For Volvox species under fixed conditions, do species differ with respect to the distribution of somatic and reproductive cell numbers in a population of asexual clones? Specifically, do they differ with respect to the dispersion of the distribution, i.e. with respect to their intrinsic variability? If so, are these differences related to major among-species developmental differences?

DATA DESCRIPTION

For each of five Volvox species, we estimate the number of somatic and reproductive cells for 40 colonies and the number of reproductive cells for an additional 200 colonies. We sampled all colonies from growing, low-density, asexual populations under standard conditions.

SEARCH METHOD

We compare the distribution of reproductive cell numbers to a Poisson distribution. We also compare the overall dispersion of reproductive cell number among species by calculating the coefficient of variation (CV). We compare the bivariate (reproductive and somatic cell) dataset to simulated datasets produced from a simple model of cell-type specification with intrinsic variability and colony size variation. This allows us to roughly estimate the level of intrinsic variability that is most consistent with our observed bivariate data (given an unknown level of size variation).

CONCLUSIONS

The overall variability (CV) in reproductive cell number is high in Volvox compared with more complex organisms. Volvox species show differences in reproductive cell number CV that were not clearly related to development, as currently understood. If we used the bivariate data and tried to account for the effects of colony size variation, we found that the species that have fast embryonic divisions and asymmetric divisions have substantially higher intrinsic variability than the species that have slow divisions and no asymmetric divisions. Under our culture conditions, the Poisson distribution is a good description of intrinsic variability in reproductive cell number for some but not all Volvox species.

The role of host population heterogeneity in the evolution of virulence

I examine here the effects of host heterogeneity in the growth of immune response on the evolution and co-evolution of virulence. The analysis is based on an extension of the 'nested model' by Gilchrist and Sasaki [Modeling host-parasite coevolution, J. Theor. Biol. 218 (2002), pp. 289-308]; the criteria for host and parasite evolution, in the paradigm of adaptive dynamics, for that model are derived in generality. Host heterogeneity is assumed to be fixed at birth according to a lognormal distribution or to the presence of two discrete types. In both cases, it is found that host heterogeneity determines a dramatic decrease in pathogen virulence, since pathogens will tune to the 'weakest' hosts. Finally we clarify how contrasting results present in the literature are due to different modelling assumptions.

Origins of Taylor's power law for fluctuation scaling in complex systems

Taylor's fluctuation scaling (FS) has been observed in many natural and man-made systems revealing an amazing universality of the law. Here, we give a reliable explanation for the origins and abundance of Taylor's FS in different complex systems. The universality of our approach is validated against real world data ranging from bird and insect populations through human chromosomes and traffic intensity in transportation networks to stock market dynamics. Using fundamental principles of statistical physics (both equilibrium and nonequilibrium) we prove that Taylor's law results from the well-defined number of states of a system characterized by the same value of a macroscopic parameter (i.e., the number of birds observed in a given area, traffic intensity measured as a number of cars passing trough a given observation point or daily activity in the stock market measured in millions of dollars).

Inferring Drosophila gap gene regulatory network: a parameter sensitivity and perturbation analysis

BACKGROUND

Inverse modelling of gene regulatory networks (GRNs) capable of simulating continuous spatio-temporal biological processes requires accurate data and a good description of the system. If quantitative relations between genes cannot be extracted from direct measurements, an efficient method to estimate the unknown parameters is mandatory. A model that has been proposed to simulate spatio-temporal gene expression patterns is the connectionist model. This method describes the quantitative dynamics of a regulatory network in space. The model parameters are estimated by means of model-fitting algorithms. The gene interactions are identified without making any prior assumptions concerning the network connectivity. As a result, the inverse modelling might lead to multiple circuits showing the same quantitative behaviour and it is not possible to identify one optimal circuit. Consequently, it is important to address the quality of the circuits in terms of model robustness.

RESULTS

Here we investigate the sensitivity and robustness of circuits obtained from reverse engineering a model capable of simulating measured gene expression patterns. As a case study we use the early gap gene segmentation mechanism in Drosophila melanogaster. We consider the limitations of the connectionist model used to describe GRN Inferred from spatio-temporal gene expression. We address the problem of circuit discrimination, where the selection criterion within the optimization technique is based of the least square minimization on the error between data and simulated results.

CONCLUSION

Parameter sensitivity analysis allows one to discriminate between circuits having significant parameter and qualitative differences but exhibiting the same quantitative pattern. Furthermore, we show that using a stochastic model derived from a deterministic solution, one can introduce fluctuations within the model to analyze the circuits' robustness. Ultimately, we show that there is a close relation between circuit sensitivity and robustness to fluctuation, and that circuit robustness is rather modular than global. The current study shows that reverse engineering of GRNs should not only focus on estimating parameters by minimizing the difference between observation and simulation but also on other model properties. Our study suggests that multi-objective optimization based on robustness and sensitivity analysis has to be considered.

The role of log-normal dynamics in the evolution of biochemical pathways

The study of the scale-free topology in non-biological and biological networks and the dynamics that can explain this fascinating property of complex systems have captured the attention of the scientific community in the last years. Here, we analyze the biochemical pathways of three organisms (Methanococcus jannaschii, Escherichia coli, Saccharomyces cerevisiae) which are representatives of the main kingdoms Archaea, Bacteria and Eukaryotes during the course of the biological evolution. We can consider two complementary representations of the biochemical pathways: the enzymes network and the chemical compounds network. In this article, we propose a stochastic model that explains that the scale-free topology with exponent in the vicinity of gamma approximately 3/2 found across these three organisms is governed by the log-normal dynamics in the evolution of the enzymes network. Precisely, the fluctuations of the connectivity degree of enzymes in the biochemical pathways between evolutionary distant organisms follow the same conserved dynamical principle, which in the end is the origin of the stationary scale-free distribution observed among species, from Archaea to Eukaryotes. In particular, the log-normal dynamics guarantees the conservation of the scale-free distribution in evolving networks. Furthermore, the log-normal dynamics also gives a possible explanation for the restricted range of observed exponents gamma in the scale-free networks (i.e., gamma > or = 3/2). Finally, our model is also applied to the chemical compounds network of biochemical pathways and the Internet network.

Quantum algorithm for programmed cell death of Caenorhabditis elegans

During the development of Caenorhabditis elegans, through cell divisions, a total of exactly 1090 cells are generated, 131 of which undergo programmed cell death (PCD) to result in an adult organism comprising 959 cells. Of those 131, exactly 113 undergo PCD during embryogenesis, subdivided across the cell lineages in the following fashion: 98 for AB lineage; 14 for MS lineage; and 1 for C lineage. Is there a law underlying these numbers, and if there is, what could it be? Here we wish to show that the count of the cells undergoing PCD complies with the cipher laws related to the algorithms of Shor and of Grover.